Model Theory and Differential Equations

本项目以代数微分方程式和模型理论为中心。微分方程式是现代数学及其在其他科学中的应用的核心。模型理论是研究被称为可定义集合的对象的数理逻辑的一部分。可定义集合的概念是灵活的，随着一个人在不同的数学环境中工作而变化。在微分代数领域，可定义集与微分方程紧密相连，提供了两个学科之间的联系。该项目计划利用这种联系来促进这两个领域的知识。该项目还寻求在其他数学领域的应用，主要是数论。这项研究项目解决了场与算符的模型理论中的问题。首先，研究人员将推广他在j-函数的微分代数上的工作，以理解与各种Shimura变种相关的解析覆盖映射所满足的微分方程。这项工作预计会产生与特殊点猜想相关的数论结果，就像它在模曲线上所做的那样。该项目还将研究微分代数中的几个有限结果；这项工作预计将对数论和计算微分代数的有效界产生影响。该项目旨在发展超单和超稳定群的模型理论的各个方面，作为最近在微分代数和差分代数群中工作的推广。该项目将研究微分闭域中分解强极小集的分类。这项工作试图回答关于微分方程的一个基本问题：固定微分方程解之间的微分代数关系给出了什么可能的结构？最后，这个项目的目的是改编和推广代数叶层和矢量场的结果，以应用于微分代数几何。

This project centers on algebraic differential equations and model theory. Differential equations are at the heart of modern mathematics and its applications in other sciences. Model theory is a part of mathematical logic in which objects known as definable sets are studied. The notion of a definable set is flexible, changing as one works in different mathematical settings. In the area of differential algebra, definable sets are closely linked with differential equations, providing a connection between the two subjects. This project plans to exploit that link to advance the knowledge of both fields. The project also seeks applications in other areas of mathematics, primarily number theory.This research project addresses problems in the model theory of fields with operators. First, the investigator will generalize his work on the differential algebra of the j-function to understand the differential equations satisfied by analytic covering maps associated with various Shimura varieties. This line of work is expected to have number theoretic consequences related to special-point conjectures as it did in the case of modular curves. The project will also investigate several finiteness results in differential algebra; this line of work is expected to have consequences on effective bounds in number theory and in computational differential algebra. The project aims to develop various aspects of the model theory of supersimple and superstable groups as a generalization of recent work in differential algebraic and difference algebraic groups. The project will investigate the classification of disintegrated strongly minimal sets in differentially closed fields. This work seeks to answer a fundamental question about differential equations: what are the possible structures given by the differential algebraic relations between solutions to a fixed differential equation? Finally, this project aims to adapt and generalize results from algebraic foliations and vector fields for application in differential algebraic geometry.